Groups and Group Convolutions

Coincidentally, I'm reading Roy McWeeny's Classic book Symmetry: An Introduction to Group Theory and Its Applications [0] right now. It's a "leisurely but reasonably thorough" (as its author describes it) look at finite groups with an eye towards scientific applications, a good practical introduction for readers with a decent math background.

Hermann Weyl's book Symmetry [1] is a more truly leisurely introduction, much more a discussion of the ideas than a math book.

0. http://www.amazon.com/gp/product/0486421821/

1. http://www.amazon.com/Symmetry-Hermann-Weyl/dp/0691023743

Although I'm a physicist group theory was one of my favorite undergraduate courses. At the risk of sounding like an idiot to a mathematician, I will attempt to explain one of the cooler parts of an extremely cool subject. There are finite and infinite groups. An example of the former is the modulo3 addition operator over the elements [0,1,2]. The identity element is 0, 0+0=0, 0+1=1, 0+2=2. The inverse of 0 is 0, of 1 is 2 (1 + 2 mod 3 = 0) and of 2 is 1 (2 + 1 mod 3 = 0). Addition is also associative so we have that covered. An example of an infinite group would be addition over the integers.

Anyway, focusing on finite groups, mathematicians spent many years trying to classify finite groups into different types. For instance, every finite group of prime length is a cyclic group (http://groupprops.subwiki.org/wiki/Group_of_prime_order). This is already very cool since it implies that for each prime number, there is exactly one group with that number of elements, anything that looks like another group can be mapped 1:1 with the cyclic group.

There are many other types of finite groups and the vast majority (this is an understatement) can be classified as one type or another. At some point mathematicians realized that although there are an infinite number of finite groups, maybe they could classify all finite groups into one type or another and set about doing so, with the caveat that there is a finite number of exceptions to this classification, and these exceptions are called sporadic groups (http://en.wikipedia.org/wiki/Sporadic_group).

Here is where my knowledge is the fuzziest but also where things get awesome. Using a form of magic only available to mathematicians, it was predicted that an extremely large sporadic group existed somewhere out there, avoiding classification into one of the standard group types. I am guessing that certain hints to its existence were popping up here and there but have no idea as to what those hints might look like. In physics, particles have been predicted before (and at least one time using group theory: http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29), these predictions usually stem from attempting to explain experimental results. Mathematics lacks these experimental results so these sorts of predictions are all the more impressive.

Anyway, back to this extremely large group. It was given the name the Monster Group and an upper bound was given as to its size. If I recall correctly (I cannot find a citation), this upper bound was considered the largest useful number of all time (any child can name a number larger than a number given to them, but using it in some novel way is not so easy). Eventually the Monster Group was found, fulfilling the prediction, and its size given as 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, far below the upper bound but still quite large nonetheless (http://en.wikipedia.org/wiki/Monster_group).

Also, mathematicians succeeded in classifying finite groups (http://en.wikipedia.org/wiki/Classification_of_finite_simple...), it only took several hundred articles and tens of thousands of pages to do so.